Trent at the newly revived Rochester philosophy blog has a couple of recent posts on pedagogy, describing how he introduces modal operators to students and how to respond to students unimpressed by the constraints of logic (for the latter problem, I recommend Gensler's strategy!). Sharing such pedagogical suggestions strikes me as a really helpful use of blogs. I probably won't be doing much (if any) teaching for another couple of years yet, but I imagine such tips could come in useful, and I find them interesting anyhow. There was once a whole blog (Metatome) dedicated to philosophical pedagogy, but it sadly seems to have died a silent death.
Anyway, I'd just like to invite all the philosophy tutors and lecturers out there to share their tricks of the trade. Feel free to post them in comments here, or on your own blog, or try to bring Metatome back to life...
BTW, one other neat example I can point to is Uriah's old post on outfoxing plagiarists. Very clever.
Friday, January 27, 2006
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On teaching the Law of Non-Contradiction: Here is a way that might satisfy almost everyone, or at least leave some room for compromise.
ReplyDeleteGet a piece of paper and draw a rectangle in the middle of it, and divide the rectangle in two. Give the student a pen and say "make a mark on the piece of paper in front
of you." Now, if they make a mark outside the rectangle, say "you are not allowed to make any marks outside the rectangle. If they ask why not?, say "because the rectangle is all there is." If they make a mark on any of the lines, say "you are not allowed to make marks on the lines", and if they question this, say "it is because the lines are infinitely small." If they make a mark that is bigger than one of the two parts of the rectangle, say "the mark must fit inside one of the two squares", and if they are still skeptical, say "look, there is not one square, but two, so you must put the mark in just one square." If the student points out that there are an infinite number of marks that one could make, in each square, and that consequently it is odd to say 'there are only two places to put the mark'", the teacher can say: "we will consider all marks in a square to be exactly the same mark."
By this time the student is well-taught, and when they are asked to put a mark on the page, they will put it in one and only one of the two squares. Now the teacher is ready to explain the law on non-contradiction, and so he says: "Now, look at this rectangle. If you are asked to make one mark on this page, there are only two things that can happen, either you put it in the left square, or you put in the right square. Note that, if you put the mark in the left square, you cannot put it in the right, and if you put it in the left, you cannot put it in the right. Now, if we let P represent the left square, and not-P represent the left square, it is perfectly obvious that P implies non-not-P. That is the Law of Non-Contradiction. It governs everything, and without it we cannot think. This is why we say 'if something is equal to two, then it cannot possibly be equal to three or four, or anything other natural number except two.' It is also why we say 'if the sky is blue, then it cannot be red.'"
This should almost satisfy both sides, because the sceptical student is free to say:
"But you have missed out a step. When you talk about possible states of the sky, you assume that the possible states of the sky are like/can be modelled on/can be usefully compared to, the rectangle on the page. This comparison might be quite convenient in lots of situations, but you should remember firstly that it is only a comparison, and secondly that there are other comparisons we could make."
And then the logician can say:
"But most of what I do has nothing to do with real sky and real colours. It is about numbers and symbols and shapes and things like that, things you can easily draw on a page. Also, we need at least some things that everyone can agree on, and statements that are precise, and in which everyone knows what we mean. The exercise on the page is an example of one of these. Also, the model we create in the case of Non-Contradiction is a special sort of model, because we can use it in almost any situation. This means that, if we derive further results from the Law of Non-Contradiction, those results can be applied in many many situations, which saves a lot of work."
Question: Would this tutorial satisfy a logician? ie. would a logician say "the exercise involving the two squares and the marks is just a pedagogical aid; the law of non-contradiction expresses something much more fundamental about the nature of thought and the nature of the world." Or would they say: "the exercise on the page, and other exercises like it (with circles instead of rectangles, with lines etched in stone instead of drawn on paper etc.) are really all there is to the law of non-contradiction, and the statement "P implies non-not-P" is just a shorthand for referring to all these exercises, in the same way that the statement "line over line gives a cross” is a quick way of referring to all those situations where we place one line perpendicularly over another line of similar length, to get a cross.”?
To follow up on Derek's point, I often deal with silences both by waiting, but then also the use of light humour, I particularly like saying, "you know I can be silent for longer than you can" I then follow that up with, "and I'm paid to be here, and you're paying me..." never fails to get a laugh.
ReplyDeleteHowever Silences might also mean that the students are overwhelmed by the question, and haven't clearly got it, or what they are being asked, so I usually ask them if this is the case.
It might also mean the opposite. I recall one class where the lecturer would ask painfully obvious questions, and be met with silence. Everyone knew the answer, and also knew that everyone else knew the answer, and so didn't feel that they had anything special to contribute. (That was my own reaction at least. I guess I can't speak for the others.)
ReplyDelete